Dictatorial-certificate polymorphisms

Extend the classification of simple polymorphisms to dictatorial-certificate polymorphisms of predicates over finite product domains, where a subset of coordinate functions depends on at most one input coordinate and their outputs form a certificate for the predicate.

Background

The paper’s notion of triviality classifies polymorphisms as either dictatorial or certificate type. It observes that with four or more candidates, another simple form arises: a subset of coordinate functions may each depend on at most one input coordinate, potentially on different coordinates, while their outputs collectively constitute a certificate for the predicate.

The authors explicitly propose extending their results to this broader class of polymorphisms, which would generalize the paper’s classification beyond the dictatorial and certificate types treated in the main theorems.